Ars Mathematica

I learned many new things from James Iry’s brief history of programming languages. For example, while I’ve used Lisp for some time now, I had no idea of how it all began:

John McCarthy and Paul Graham invent LISP. Due to high costs caused by a post-war depletion of the strategic parentheses reserve LISP never becomes popular. In spite of its lack of popularity, LISP (now “Lisp” or sometimes “Arc”) remains an influential language in “key algorithmic techniques such as recursion and condescension”

Condescension has never made my programs run faster, but it’s what makes writing them worthwhile.

I was looking for an introduction to the topic of random matrices, and I came across this survey article by Edelman and Rao on the subject. It considers a somewhat broader point of view than just results on the random distribution of eigenvalues, which are the most famous results in the subject.

One thing I found interesting is that you can explicitly calculate the Jacobian of various matrix decompositions as nonlinear functions of the matrix entries. They use this to help explain results on the random distribution of eigenvalues. More on this approach can be found in Edelman’s thesis.

This is the 5th year anniversary of this blog. The list of things I’ve managed to do for 5 years is very short, but this is one of the things on it.

The investment bank Goldman Sachs is being sued by the SEC for allegedly selling an investment designed to lose money. The investment was built on a pool of mortgages that were likely to go into default. Initially, Erik Gerding at The Conglomerate (a legal blog) thought that the SEC would have difficulty winning the case, since Goldman had disclosed the contents of the pool. Then he had second thoughts, because of this paper, “Computational Complexity and Information Asymmetry in Financial Products”, by Arora, Barak, Brunnermeier, and Ge. The paper shows that even if you know the contents of the pool, detecting whether bad mortgages are hidden in the pool is an NP-complete problem, which is normally considered the hallmark of computational intractability.

In some disciplines, there is the notion of a Comment on a published article, which is what it sounds like: a short comment about the contents of the article (for example, that it’s wrong). Cat Dynamics links to an interesting account of physicist Rick Trebino’s (lengthy but ultimately successful) attempts to publish a Comment explaining why a published article is wrong.

I don’t think I’ve ever seen a Comment in a pure math journal. They’re common in statistics journals.

The New York Times has an online math blog by Steven Strogatz. Given the venue, it is written for an elementary audience. Here is a recent post on the method of exhaustion and how it allows you to approximate π.

The Insane Clown Posse has declared war on scientists. Check out these recent lyrics (warning, curse words ahead):

Read the rest of this entry »

When I said that I was about to eat a big bowl of non-blogging, I didn’t mean that I was going to turn off the website, but apparently that’s what the server thought I meant. I didn’t realize initially that the site was down because I was having unrelated DNS issues, and I thought that’s why I couldn’t access the site.

I discovered that the site was back up when I received my first notification that somebody had posted spam in the comment section.

This article by Henrik Lenstra has an intriguing quote:

The key notion underlying the second algorithm is that of “infrastructure”, a word coined by Shanks (see [11]) to describe a certain multiplicative structure that he detected within the period of the continued fraction expansion of √d. It was subsequently shown (see [7]) that this period can be “embedded” in a circle group of “circumference” Rd, the embedding preserving the cyclical structure. In the modern terminology of Arakelov theory, one may describe that circle group as the kernel of the natural map Pic0Z[√d] → PicZ[√d] from the group of “metrized line bundles of degree 0” on the “arithmetic curve” corresponding to Z[√d] to the usual class group of invertible ideals. By means of Gauss’s reduced binary quadratic forms one can do explicit computations in Pic0Z[√d] and in its “circle” subgroup.

I’m a sucker for anything that related elementary topics (like Pell’s equation) to advanced topics that I don’t understand (like Arakelov theory).